Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant part $V^G$ of the representation.

Now let $G = GL(1)$ (or $\mathbb{G}_m$ or $\mathbb{C}^\ast$ if you like). Then let $\phi_n : \mathbb{P}^n \to BG$ be the map corresponding to the bundle $\mathcal{O}(1)$ (or maybe we should take map corresponding to the bundle $\mathcal{O}(-1)$, I always mix it up).

Let $V$ be a representation of $G$. Then we can compute the index (i.e. sheaf cohomology Euler characteristic) of $\phi_n^\ast V$ over $\mathbb{P}^n$. We can show, just by doing the calculation, that for sufficiently large $n$ (how large depends on the representation $V$), this index is a polynomial in $n$. Then, plugging in $n=-1$ into this polynomial, one can show that the result is equal to $\chi(BG; V)$, defined as above.

So, in this fashion, we can recover the index over the stack $BGL(1)$ if we know the index over each $\mathbb{P}^n$ (for sufficiently large $n$).

This result seems to make sense, because at least in topology we have $B\mathbb{C}^\ast \simeq \mathbb{CP}^\infty$, and at least intuitively we can think of $\mathbb{CP}^n$ as "approximating" $\mathbb{CP}^\infty$.

But so far I don't really have a satisfactory explanation for this, other than "it follows by doing the computation" and "it seems to make intuitive sense" as I've explained above. I wonder if there is a better way to see this; does it follow by some deeper facts?; does it follow by some more general theorems? I'm sorry that my question is not very precise.

I would also be interested in any other theorems or results which relate finite dimensional projective spaces with $BGL(1)$. (There are already a few such results in the link in S. Carnahan's comment below.)

I don't think this is enough to be an answer, but comments 23 (by David Ben-Zvi) and 27 (by "bb") in the following blog post are especially informative: sbseminar.wordpress.com/2008/06/19/whats-a-stack In particular, $\mathbb{A}^1$ homotopy theory gives one rigorous sense in which the classifying stack of $GL(1)$ is approximated by projective spaces: one has a sequence of morphisms whose fibers are increasingly connected.
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S. Carnahan♦May 23 '11 at 5:09

I've added a bounty in an attempt to attract more answers...
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Kevin H. LinMay 25 '11 at 22:40

1 Answer
1

The paper arXiv:0808.2785 of Anderson, Griffeth, and Miller uses precisely this approximation technique to prove a certain positivity result in the torus equivariant k-theory of homogeneous spaces (see their prop. 3.1). AGR reference the paper